2.8 Complex Zeros & Fund. Th. of Algebra (KEY)
Remember to show your work on your assignment where necessary!
1. zeros: x = 3i, x = -3i, f ( x)  x2  9 , no x-intercepts
2. zeros: x = 1 mult. 2, x = 2i, x = -2i, f ( x)  x4  2 x3  5x2  8x  4 , x-int. (1, 0)
3. f ( x)   x 1  2i  x 1  2i  , degree 2, no x-intercepts
4.
f ( x)   x  2 x  3  x  i  x  i  , degree 4, x-intercepts (2, 0) (3, 0)
5. f ( x)   x  2 x 1  2i  x 1 2i  , degree 3, x-intercepts (-2, 0)
6.



f ( x)  x  3 x  3  x  4 x  5  6i  x  5  6i  , degree 5, x-intercepts (-4, 0)


3,0  3,0

7. f ( x)  x5  2 x4 10x3  20x2  9x 18 , degree 5, x-int. (2, 0)
8. f ( x)  x3  4 x2  29 x , degree 3, x-int. (0, 0)
9.
f ( x)   x 1  x  2
2
10. f ( x)   x  2
11.
3
2
, degree 5, x-int. (1, 0) (-2, 0)
 x  3  i  x  3  i  , degree 4, x-int. (2, 0)
f ( x)   x4  6 x3  9 x2 , degree 4, x-int. (0, 0) (3, 0)
12 – 15 on your own!

1
19 
1
19 
1
1
19
19
16. zeros: x = 1, x =  
i 
x


i 
i, x=  
i : f ( x)   x 1  x  

2
2
2
2
2
2
2
2




1
3 
1
3 
1
3
1
3
x


i
i , x =   i : f ( x)   x  2 3x 1  x   i 
17. zeros: x = -2, x = 1/3, x = - 
2 2 
2 2 
2 2
2 2








18. remaining zeros: x = 1 - i, x =
3 , x = - 3 : f ( x)  x  3 x  3  x 1  i  x 1  i 
19. remaining zeros: x = -4 i, x =
3 , x = - 3 : f ( x)  x  3 x  3  x  4i  x  4i 


20. f ( x)   x 1 2 x2  x  3


21. f ( x)  x2  4  x  1 x  3
22-23. On your own! Bonus: on you own!